## THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS

Citations: | 5 - 0 self |

### BibTeX

@MISC{Kitchloo_thomprospectra,

author = {Nitu Kitchloo and Jack Morava},

title = {THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS},

year = {}

}

### OpenURL

### Abstract

Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new localization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.

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Citation Context ...sion 1 face ∆i for 1 ≤ i ≤ n is identified with the subset of the alcove {(1, h) | αi(h) = 0}, for i ̸= 0, and ∆0 is identified with the subspace {(1, h) | α0(h) = 1}. General facts about Loop groups =-=[22, 26]-=- show that the surjective map LalgG × ∆ −→ A, (f(z), y) ↦→ Ad f(z)(y) has isotropy HI on the subspace ∆I. Hence it factors through a T˜×LalgG-equivariant homeomorphism between A(LalgG) and the affine ... |

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Citation Context ... of their results. Recall also that A(LG) is the classifying space for proper actions (i.e. with compact isotropy) so these results also bear an interesting relationship to the Baum-Connes conjecture =-=[7]-=- Question. For the manifold LM, with frame bundle LP, we can construct a spectrum DLM := holimI LP+ ∧HI Ad(HI)+ It would be very interesting to understand something about KT(DLM). 4. Localization Theo... |

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Citation Context ...a point defines a kind of fundamental class, and the dual to the diagonal of M makes M −TM into a ring-spectrum. The ring structure of M −TM has been studied by various authors (see for example [13], =-=[25]-=-). Prospectus: Chas and Sullivan [11] have constructed a very interesting product on the homology of a free loopspace, suitably desuspended, motivated by string theory. Cohen and Jones [15] saw that t... |

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Citation Context ...ver, there is more to our construction than a simple homotopy functor: it comes with a natural (fixed-point) orientation, which defines a systematic theory of Thom isomorphisms. In the terminology of =-=[4]-=-, it is represented by an elliptic spectrum, associated to the Tate curve over Z((q)); its natural orientation is defined by the σ-function of §4. Remark 5.7. Let HT denote T-equivariant singular Bore... |

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Citation Context ...s easy to check from this that the cokernel for nontrivial powers of z is isomorphic to the cokernel of ϕ0 restricted to u −1 Z[u + u −1 , z] and hence is ⊕ k≥0 which agrees with the classical result =-=[18]-=-. Z[u + u −1 ] 〈Sym k+1 (u + u −1 )〉 (z/u)k+2 We now get to the collapse of the spectral sequence. It is sufficient to establish: Proposition 3.6. Assume n > 0, then colim i I Kr+1,n HI (Ad(HI)+) is t... |

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Citation Context ... PROSPECTRA FOR LOOPGROUP REPRESENTATIONS 9 A conjectural relationship with the work of Freed, Hopkins and Teleman The discussion below assumes the existence of LG-equivariant K-theory, as defined in =-=[19]-=- (see Appendix). Given a space X with a proper LG-action, Freed-HopkinsTeleman define an LG-equivariant spectrum over X. The equivariant K-theory groups are defined as the homotopy groups of the space... |

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Citation Context ...e that its underlying non-equivariant object maps naturally to (a thickening of) the Chas-Sullivan spectrum. 3. The dualizing spectrum of LG The dualizing spectrum of a topological group K is defined =-=[24]-=- as the K-homotopy fixed point spectrum: DK = K hK + = F(EK+, K+) K where K+ is the suspension spectrum of the space K+, endowed with a right Kaction. The dualizing spectrum DK admits a K-action given... |

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Citation Context ... completion of a much smaller (elliptic) cohomology theory, whose coefficients are modular forms, with the completion map corresponding to the q-expansion. The geometry underlying modularity is still =-=[10]-=- quite mysterious. Remark 5.6. A cohomology theory defined on finite spectra extends to a cohomology theory on all spectra [1]; moreover, any two extensions are equivalent, and the equivalence is uniq... |

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Citation Context ... by the orientation-reversing involution λ ↦→ λ −1 of T. It is in some sense a chiral completion. Remark 5.9. The completion theorem above is a specialization of Segal’s original localization theorem =-=[29]-=-, which says that KT(X), considered as a sheaf over the multiplicative groupscheme Spec KT = Gm (cf. [28]), has for its stalks over generic (ie nontorsion) points, the K-theory of the fixed point spac... |

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Citation Context ...ere first studied by Quillen, and were explored further by S. Mitchell [26]. The first author has studied these buildingsTHOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS 5 for a general Kac-Moody group =-=[23]-=-; most of the properties of the affine building used below hold for this larger class. Theorem 2.2. There exist a certain finite set of compact ‘parabolic’ subgroups HI of LG (see 7.2), such that the ... |

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Citation Context ...5.9. The completion theorem above is a specialization of Segal’s original localization theorem [29], which says that KT(X), considered as a sheaf over the multiplicative groupscheme Spec KT = Gm (cf. =-=[28]-=-), has for its stalks over generic (ie nontorsion) points, the K-theory of the fixed point space. The Tate point Spec Z((q)) → Spec Z[q ± ] is an example of such a generic point, perhaps too close to ... |

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Citation Context ... fixed-point orientation defined above will have good multiplicative properties. Such Weierstrass products sometimes behave better when ‘renormalized’, by dividing by their values on constant bundles =-=[2]-=-. If E is KT with the usual complex (Todd) orientation, we have e(L) +K [k](q) = 1 − q k L ; but for our purposes things turn out better with the Atiyah-Bott-Shapiro spin orientation; in that case the... |

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Citation Context ... A(LG) −U = hocolimI LG+ ∧HI S −U I , where LG+ denotes LG, with a disjoint basepoint. Homotopy colimits in the category of prospectra can be defined in general, using the model category structure of =-=[12]-=-. Remark 2.9. Given any principal LG-bundle E → B, and a representation U of LG, we define the Thom prospectrum of the virtual bundle associated to the representation −U to be B −U ! = E+ ∧LG A(LG) −U... |

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Citation Context ... maps. For the case at hand, it is clear that this cohomology theory is equivalent to the formal extension K((q)), where K is complex K-theory, with q a parameter in degree zero. Hovey and Strickland =-=[20]-=- have shown that an evenly graded spectrum does not support phantom maps, so our cohomology theory is uniquely equivalent, as a homotopy functor, to K((q)). However, there is more to our construction ... |

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Citation Context ...opspaces to cohomology theories on the fixedpoints, with related (but distinct) formal groups, is part of an emerging understanding of what homotopy theorists call ‘chromatic redshift’ phenomena, cf. =-=[2, 3, 31]-=-.THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS 17 Remark 5.8. Our completion of equivariant K-theory is the natural repository for characters of positive-energy representations of loop groups; it is ... |

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Citation Context ...(1 − q k L)(1 − q k L −1 ) (1 − q k ) 2 for the Weierstrass sigma-function as Thom class for a line bundle L. This extends by the splitting principle to define the orientation giving the Witten genus =-=[32]-=-. 5. One moral of the story Since the early 80’s physicists have been trying to interpret M ↦→ KT(LM) as a kind of elliptic cohomology theory; but of course we know better, because we know that mappin... |

1 |
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Citation Context ...opspaces to cohomology theories on the fixedpoints, with related (but distinct) formal groups, is part of an emerging understanding of what homotopy theorists call ‘chromatic redshift’ phenomena, cf. =-=[2, 3, 31]-=-.THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS 17 Remark 5.8. Our completion of equivariant K-theory is the natural repository for characters of positive-energy representations of loop groups; it is ... |

1 |
String topology, available at math.AT/9911159
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Citation Context ... class, and the dual to the diagonal of M makes M −TM into a ring-spectrum. The ring structure of M −TM has been studied by various authors (see for example [13], [25]). Prospectus: Chas and Sullivan =-=[11]-=- have constructed a very interesting product on the homology of a free loopspace, suitably desuspended, motivated by string theory. Cohen and Jones [15] saw that this product comes from a ring-spectru... |

1 |
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Citation Context ...on to a point defines a kind of fundamental class, and the dual to the diagonal of M makes M −TM into a ring-spectrum. The ring structure of M −TM has been studied by various authors (see for example =-=[13]-=-, [25]). Prospectus: Chas and Sullivan [11] have constructed a very interesting product on the homology of a free loopspace, suitably desuspended, motivated by string theory. Cohen and Jones [15] saw ... |

1 |
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Citation Context ...ine space A(S 1 × G) implies that the fixed point space A(LG) K is contractible for any compact subgroup K ⊆ T˜×LG. If E → B is a principal bundle with structure group LG, then (motivated by ideas of =-=[14]-=-) we construct a ‘thickening’ of B: Definition 2.5. The thickening of B associated to the bundle E is the space B † (E) = E ×LG A(LG) = hocolimI E/HI . We will omit E from the notation, when the defin... |

1 |
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Citation Context ... the equivariant K-theory of our construction, and we show how this recovers the Witten genus from a string-topological point of view. We plan to discuss actions of various string-topological operads =-=[15]-=- on our construction in a later paper; that work is in progress. We would like to thank R. Cohen, V. Godin, S. Stolz, and A. Stacey for many helpful conversations, and we would also like to acknowledg... |

1 |
Floer’s infinite-dimensional Morse theory and homotopy theory, in The Floer memorial volume
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Citation Context ...ill make it possible to say something more explict about this.4 NITU KITCHLOO AND JACK MORAVA 2. Problems & Solutions For our constructions, we need two pieces of technology: Cohen, Jones, and Segal =-=[16]-=-(appendix) associate to a filtration E : · · · ⊂ Ei ⊂ Ei+1 ⊂ . . . of an infinite-dimensional vector bundle over X, a pro-object X −E : · · · → X −Ei+1 → X −Ei → . . . in the category of spectra. [A r... |

1 |
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Citation Context ...cs Subject Classification. 22E6, 55N34, 55P35. NK is partially supported by NSF grant DMS 0436600, JM by DMS 0406461. 12 NITU KITCHLOO AND JACK MORAVA unfamiliar, and has been difficult to work with =-=[17]-=-. The main conceptual result of this note [which was motivated by ideas of Cohen, Godin, and Segal] is the definition of a canonical ‘thickening’ L † M of a free loopspace, with the same equivariant h... |

1 |
theorem on buildings and the loops on a symmetric space, Enseign
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Citation Context ...LG. The main step toward our resolution of this problem depends on the following result, proved in §7 below. Such constructions were first studied by Quillen, and were explored further by S. Mitchell =-=[26]-=-. The first author has studied these buildingsTHOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS 5 for a general Kac-Moody group [23]; most of the properties of the affine building used below hold for thi... |

1 | Forms of K-theory, Math Zeits - Morava - 1989 |

1 |
What is an elliptic object? available at math.ucsd.edu
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Citation Context ...g = 0 this is the Witten genus of M, and when g = 1 it is the Euler characteristic. Finally: our construction is, from its beginnings onward, formulated in terms of closed strings. Stolz and Teichner =-=[30]-=- have produced a deeper approach to a theory of elliptic objects, which promises to incorporate interesting aspects of open strings as well. However, their theory is in some ways quite complicated; an... |